Problem: Find the remainder when the sum \[75+76+77+78+79+80+81+82\]is divided by 16.
Answer: We notice that 16 divides $78+82$ as well as $79+81$ and also 80.  Therefore the sum is congruent to  \[75+76+77\pmod{16}.\]Since these numbers are congruent to $-5$, $-4$, and $-3$ modulo 16, this can be computed as  \[-5-4-3\equiv-12\pmod{16}.\]Finally, since $-12\equiv4\pmod{16}$ the remainder we seek is $\boxed{4}$.